Advances in Mathematics· 2026Q1
On Smirnov's approach to the abc conjecture
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- 2026year
Short summary
Algebraic geometry over pointed monoids provides an intrinsic interpretation for Alexander Smirnov's compactification of the spectrum of the ring of integers of a number field K, crucial for his ABC conjecture approach.
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Key points
- Applies algebraic geometry over pointed monoids to interpret Smirnov's compactification.
- Provides an intrinsic geometric meaning for the compactification of the spectrum of the ring of integers of a number field K.
- Explains maps between these compactified structures induced by elements of K.
- Connects these geometric interpretations to Alexander Smirnov's approach to the ABC conjecture.
AI-generated from the title and abstract; the full text is not read.
Abstract
We use algebraic geometry over pointed monoids to give an intrinsic interpretation for the compactification of the spectrum of the ring of integers of a number field $K$, for the projective line over algebraic extensions of $\mathbb{F}_1$ and for maps between them induced by elements of $K$, as introduced by Alexander Smirnov in his approach to the ABC conjecture.
The authors' abstract, as published at the source. Advances in Mathematics, 2026 · DOI ↗
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Field: Geometry and Topology
Geometry and TopologyMathematics